Question

The greatest of the numbers 1, 2^1/2, 3^1/3, 4^1/4, 5^1/5, 6^1/6 and 7^1/7 is

• A. 2^1/2
• B. 3^1/3
• C. 7^1/7
• D. All are equal

Answer 1

Answer: (B)

3^1/3

Answer 2

Assuming the function

f(x) = x^1/x {x > 0}

f '(x) = x^1/x $\left\lbrack \frac{d}{dx}\left( \frac{1}{x}\log x \right) \right\rbrack$

f '(x) = x^1/x $\left\lbrack - \frac{\log x}{x^{2}} - \frac{1}{x^{2}} \right\rbrack$

f '(x) = – $\frac{x^{1/x}}{x^{2}}$ (1 + log x)

clearly when x > e, f '(x) will be negative f(x) is decreasing

when 0 < x < e, f '(x) will be positive f(x) is increasing

it means 3^1/3 > 4^1/4 > 5^1/5 > 6^1/6 > 7^1/7

Now compare (1) and (2)^1/2 and (3)^1/3

Apply same power on given three numbers

(1)⁶ (2)^{1/2 × 6} (3)^{1/3 × 6}

1 2³ 3²

1 < 8 < 9

(1) <(2)^1/2 < (3)^1/3

So maximum number is (3)^1/3